Finite Group Databasebeta

← C2×#9517 #201861

Subgroups — C2×#9517 #201861

Order 512.

1432 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2973C211yesmaximal, characteristic
2C2×#2579C211yesmaximal, characteristic
3C2×#1465C211yesmaximal, characteristic
4C2×#1463C212yesmaximal, complemented
5#9517C218yesmaximal, direct product factor
6C2×#1305C212yesmaximal, complemented
7C22×C4×D4C2211yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9#2973—24no—
10#2973—22no—
11#2973C2214yes—
12C2×#435—24no—
13C2×#432—21no—
14C2×#407C2212yes—
15C2×#407C2212yes—
16C22×C22⋊C8—21no—
17#2579—22no—
18#2579—22no—
19#2579C414yes—
20#2579C2214yes—
21C2×#398—21no—
22C2×#396—21no—
23C2×#387—21no—
24C2×#386C411yescharacteristic
25C2×#386C411yescharacteristic
26C2×#385—21no—
27C2×#385C411yescharacteristic
28C2×#385C411yescharacteristic
29C2×#380C2211yescharacteristic
30#1465—22no—
31#1465C2214yes—
32#1463—24no—
33#1463C2218yescomplemented
34C2×#324—24no—
35C2×#324—21no—
36C2×#323—21no—
37#1305C2218yescomplemented
38C23×D4C2×C411yescharacteristic
39C24×C4C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C22×M16—22no—
44C22×M16—22no—
45C23×C8—22no—
46C2×C4×D4—22no—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4C2×C412yes—
51C2×C4×D4D412yes—
52C2×C4×D4D412yes—
53C2×C4×D4C2312yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C411yescharacteristic
73C22×C22⋊C4D411yescharacteristic
74C22×C42D411yescharacteristic
75#435—216no—
76#435—216no—
77#432—24no—
78#432—24no—
79C2×C4⋊C8—22no—
80#407—22no—
81#407—22no—
82#407—22no—
83#407—22no—
84#407—22no—
85#407D414yes—
86#407C2314yes—
87#407C2314yes—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8D414yes—
95C2×#97—22no—
96#398—24no—
97#398—24no—
98#396—24no—
99#396—24no—
100#387—24no—
101#387—24no—
102#386—22no—
103#386—22no—
104#386C2×C414yes—
105#386C2×C414yes—
106#385—24no—
107#385—24no—
108#385—22no—
109#385—22no—
110#385C2×C414yes—
111#385C2×C414yes—
112#380—21no—
113#380—21no—
114#380C2×C412yes—
115#380C2312yes—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121#324—24no—
122#324—216no—
123#323—24no—
124C25C22⋊C411yescharacteristic
125C25C22×C411yescharacteristic
126C22×D4—24no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—42no—
130C22×D4C22⋊C412yes—
131C22×D4C22×C412yes—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—44no—
147C23×C4—82no—
148C23×C4—82no—
149C23×C4—82no—
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C412yes—
154C23×C4C22×C412yes—
155C23×C4C22×C412yes—
156C23×C4C22×C411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4#4112yes—
165C23×C4#4112yes—
166C23×C4C2411yescharacteristic
167C2×M16—24no—
168C2×M16—24no—
169C2×M16—24no—
170C2×M16—42no—
171C2×M16—84no—
172C2×M16—84no—
173C22×C8—24no—
174C22×C8—28no—
175C22×C8—42no—
176C22×C8—84no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4C22⋊C412yes—
227C2×C22⋊C4C22⋊C412yes—
228C2×C22⋊C4C22×C412yes—
229C2×C22⋊C4C22×C412yes—
230C2×C22⋊C4C22×C412yes—
231C2×C22⋊C4C2×D412yes—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42C22⋊C412yes—
237C2×C42C2×D412yes—
238C4⋊C8—28no—
239C22⋊C8—28no—
240C22⋊C8—24no—
241C22⋊C8—24no—
242C22⋊C8—24no—
243C22⋊C8—24no—
244C22⋊C8—24no—
245C22⋊C8—42no—
246C22⋊C8—42no—
247C22⋊C8C2×D418yes—
248#97—28no—
249#95—22no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—22no—
255C24—21no—
256C24—22no—
257C24—22no—
258C24—21no—
259C24—22no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24#10114yes—
265C24C2×C22⋊C411yescharacteristic
266C24C2×C22⋊C411yescharacteristic
267C24C2×C22⋊C412yes—
268C24C4×D412yes—
269C24C4×D412yes—
270C24C23×C411yescharacteristic
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—48no—
278C2×D4—48no—
279C22×C4—24no—
280C22×C4—28no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—48no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—82no—
326C22×C4—81no—
327C22×C4—81no—
328C22×C4—81no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—81no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C4×D412yes—
346C22×C4C4×D412yes—
347C22×C4#12011yescharacteristic
348C22×C4#12011yescharacteristic
349C22×C4#12012yes—
350C22×C4#12112yes—
351C22×C4#12112yes—
352C22×C4#12112yes—
353C22×C4#12112yes—
354C22×C4#12311yescharacteristic
355C22×C4#12311yescharacteristic
356C22×C4#12312yes—
357C22×C4C22×D412yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C2×#4112yes—
361M16—84no—
362M16—84no—
363M16—84no—
364M16—84no—
365C2×C8—28no—
366C2×C8—48no—
367C2×C8—44no—
368C2×C8—88no—
369C2×C8—84no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—48no—
373C4⋊C4—42no—
374C4⋊C4—44no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C4⋊C4—84no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—24no—
386C22⋊C4—48no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—84no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C23—22no—
397C23—22no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—24no—
407C23—22no—
408C23—22no—
409C23—24no—
410C23—41no—
411C23—42no—
412C23—41no—
413C23—41no—
414C23—42no—
415C23—41no—
416C23—48no—
417C23—48no—
418C23—42no—
419C23—42no—
420C23—48no—
421C23—42no—
422C23—42no—
423C23—81no—
424C23—84no—
425C23—81no—
426C23#38612yes—
427C23#38611yescharacteristic
428C23#38611yescharacteristic
429C23C2×#10112yes—
430C23C2×#10114yes—
431C23#41312yes—
432C23C22×C22⋊C411yescharacteristic
433C23C2×C4×D412yes—
434D4—48no—
435D4—48no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—28no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—48no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—84no—
489C2×C4—84no—
490C2×C4—82no—
491C2×C4—84no—
492C2×C4—82no—
493C2×C4—84no—
494C2×C4#38611yescharacteristic
495C2×C4#38611yescharacteristic
496C2×C4#38612yes—
497C2×C4C2×#12011yescharacteristic
498C2×C4C2×#12112yes—
499C2×C4C2×#12311yescharacteristic
500C8—48no—
501C8—84no—
502C22—21no—
503C22—21no—
504C22—24no—
505C22—24no—
506C22—21no—
507C22—21no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—24no—
515C22—44no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—48no—
521C22—42no—
522C22—48no—
523C22—48no—
524C22—42no—
525C22—82no—
526C22—84no—
527C22#189112yes—
528C22#190112yes—
529C22#190112yes—
530C22#191312yes—
531C22C2×#38611yescharacteristic
532C22C22×#10111yescharacteristic
533C22C2×#41311yescharacteristic
534C4—21no—
535C4—22no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—22no—
540C4—42no—
541C4—44no—
542C4—42no—
543C4—42no—
544C4—82no—
545C4—82no—
546C4—82no—
547C2—21no—
548C2—21no—
549C2—22no—
550C2—42no—
551C2—42no—
552C2—44no—
553C2#951714yesdirect product factor
554C2C2×#189111yescharacteristic
555C2C2×#190111yescharacteristic
556C2C2×#191311yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C2×#1463C2semidirect
5. #9517C2direct (N × H)
6. C2×#1305C2semidirect
33. #1463C22semidirect
37. #1305C22semidirect
553. C2#9517direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#29732
C2×#25792
C2×#14652
C2×#14632
#95172
C2×#13052